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Suppose that $F(x,y,z)=0$ is an equation so that any variable can be solved in terms of the other two. Show that

$\displaystyle \frac{\partial x}{\partial y}\cdot\frac{\partial y}{\partial z}\cdot \frac{\partial z}{\partial x}= -1.$

I understand what the question means and I am able to see how it works for a linear equation but assuming I can't set up a general linear equation (I don't think it would be sufficient enough) how would I do this proof.

mfl
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user153009
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1 Answers1

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Let me write it as $$x_yy_zz_x$$ $$=x_y\dfrac{z_x}{z_y}$$ and by implicit differantation remember that $y_x=\dfrac{-G_x}{G_y}$ so we have $x_y(-y_x)=-1$.

Note: we regard $z=G(x,y)$

mesel
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